I believe it's the former, but I don't have the part to check
–
badp♦Apr 16 '13 at 18:30

I've gotten name upgrade notifications after kills rather than after domination messages. (I'm a fairly mediocre player; I rarely get a domination to begin with but my basic flamethrower's got an impressive adjective by now)
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ShadurApr 17 '13 at 9:30

"Increases if the actual domination just so happened to take place on an assist" Considering that the item description specifically states that it tracks "the number of enemies you dominate by killing them with that weapon", I doubt it.
–
BoltClockApr 17 '13 at 19:11